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## 'Into the Wilderness' printed from http://nrich.maths.org/

**To multiply complex numbers** we can expand brackets, taking care to remember that $i^2 = -1$. For example:

$$\eqalign{\left(4+2i\right)\left(\frac{7}{5}- 3i\right) &= \frac{28}{5} + \frac{14}{5}i - 12i - 6i^2 \\ &=

\left(\frac{28}{5} + 6\right) + \left(\frac{14}{5} - 12\right)i \\ &= \frac{58}{5} - \frac{46}{5}i.}$$

We have created the GeoGebra interactivity below for you to explore the questions that follow.

Pick a complex number $z_1$, eg $4 + 2i$, and a positive real number $z_2$, eg 3.

What is $z_1 z_2$?

What happens as you change $z_1$ and $z_2$?

Can you describe geometrically the effect of multiplying by a positive real number?

What if $z_2$ is negative?

Now explore the effect of multiplying complex numbers by $i$ (you could set $z_2 = i$ in the Geogebra tool). Can you describe the effect geometrically?

Now try this for multiplication by $-i$.

What about multiplication by $2i$, $-2i$, $\frac{1}{2}i$, $ki$ for real $k$?

What, geometrically, is the effect of multiplying by $1+i$?

Can you find any complex numbers where multiplying by them stretches but doesn't rotate?

Can you find any complex numbers where multiplying by them rotates but doesn't stretch?

**Can you predict the geometric effect of multiplication by the complex number $a + bi$?**
*Now that you have explored multiplication of numbers, you might like to start *Mapping the Territory.